\(\int \frac {e+f x^2}{\sqrt {a-b x^2} (c-d x^2)^{3/2}} \, dx\) [51]

Optimal result

Integrand size = 32, antiderivative size = 242 \[ \int \frac {e+f x^2}{\sqrt {a-b x^2} \left (c-d x^2\right )^{3/2}} \, dx=-\frac {(d e+c f) x \sqrt {a-b x^2}}{c (b c-a d) \sqrt {c-d x^2}}+\frac {(d e+c f) \sqrt {a-b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {1-\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {e \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {a-b x^2} \sqrt {c-d x^2}} \]

[Out]

-(c*f+d*e)*x*(-b*x^2+a)^(1/2)/c/(-a*d+b*c)/(-d*x^2+c)^(1/2)+(c*f+d*e)*EllipticE(x*d^(1/2)/c^(1/2),(b*c/a/d)^(1
/2))*(-b*x^2+a)^(1/2)*(1-d*x^2/c)^(1/2)/(-a*d+b*c)/c^(1/2)/d^(1/2)/(1-b*x^2/a)^(1/2)/(-d*x^2+c)^(1/2)+e*Ellipt
icF(x*d^(1/2)/c^(1/2),(b*c/a/d)^(1/2))*(1-b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)/c^(1/2)/d^(1/2)/(-b*x^2+a)^(1/2)/(-
d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {541, 538, 438, 437, 435, 432, 430} \[ \int \frac {e+f x^2}{\sqrt {a-b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \sqrt {1-\frac {d x^2}{c}} (c f+d e) E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {1-\frac {b x^2}{a}} \sqrt {c-d x^2} (b c-a d)}+\frac {e \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {a-b x^2} \sqrt {c-d x^2}}-\frac {x \sqrt {a-b x^2} (c f+d e)}{c \sqrt {c-d x^2} (b c-a d)} \]

[In]

Int[(e + f*x^2)/(Sqrt[a - b*x^2]*(c - d*x^2)^(3/2)),x]

[Out]

-(((d*e + c*f)*x*Sqrt[a - b*x^2])/(c*(b*c - a*d)*Sqrt[c - d*x^2])) + ((d*e + c*f)*Sqrt[a - b*x^2]*Sqrt[1 - (d*
x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], (b*c)/(a*d)])/(Sqrt[c]*Sqrt[d]*(b*c - a*d)*Sqrt[1 - (b*x^2)/a]*
Sqrt[c - d*x^2]) + (e*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], (b*c)/(a*
d)])/(Sqrt[c]*Sqrt[d]*Sqrt[a - b*x^2]*Sqrt[c - d*x^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 432

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 437

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]
, Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ
[a, 0]

Rule 438

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]
, Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]

Rule 538

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c]))))))

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {(d e+c f) x \sqrt {a-b x^2}}{c (b c-a d) \sqrt {c-d x^2}}-\frac {\int \frac {-c (b e+a f)+b (d e+c f) x^2}{\sqrt {a-b x^2} \sqrt {c-d x^2}} \, dx}{c (b c-a d)} \\ & = -\frac {(d e+c f) x \sqrt {a-b x^2}}{c (b c-a d) \sqrt {c-d x^2}}+\frac {e \int \frac {1}{\sqrt {a-b x^2} \sqrt {c-d x^2}} \, dx}{c}+\frac {(d e+c f) \int \frac {\sqrt {a-b x^2}}{\sqrt {c-d x^2}} \, dx}{c (b c-a d)} \\ & = -\frac {(d e+c f) x \sqrt {a-b x^2}}{c (b c-a d) \sqrt {c-d x^2}}+\frac {\left (e \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {1}{\sqrt {a-b x^2} \sqrt {1-\frac {d x^2}{c}}} \, dx}{c \sqrt {c-d x^2}}+\frac {\left ((d e+c f) \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {\sqrt {a-b x^2}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{c (b c-a d) \sqrt {c-d x^2}} \\ & = -\frac {(d e+c f) x \sqrt {a-b x^2}}{c (b c-a d) \sqrt {c-d x^2}}+\frac {\left ((d e+c f) \sqrt {a-b x^2} \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {\sqrt {1-\frac {b x^2}{a}}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{c (b c-a d) \sqrt {1-\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {\left (e \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}}} \, dx}{c \sqrt {a-b x^2} \sqrt {c-d x^2}} \\ & = -\frac {(d e+c f) x \sqrt {a-b x^2}}{c (b c-a d) \sqrt {c-d x^2}}+\frac {(d e+c f) \sqrt {a-b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {1-\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {e \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {a-b x^2} \sqrt {c-d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.74 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.91 \[ \int \frac {e+f x^2}{\sqrt {a-b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\frac {\sqrt {-\frac {b}{a}} d (d e+c f) x \left (a-b x^2\right )+i b c (d e+c f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c (-b c+a d) f \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} c d (-b c+a d) \sqrt {a-b x^2} \sqrt {c-d x^2}} \]

[In]

Integrate[(e + f*x^2)/(Sqrt[a - b*x^2]*(c - d*x^2)^(3/2)),x]

[Out]

(Sqrt[-(b/a)]*d*(d*e + c*f)*x*(a - b*x^2) + I*b*c*(d*e + c*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*Elliptic
E[I*ArcSinh[Sqrt[-(b/a)]*x], (a*d)/(b*c)] + I*c*(-(b*c) + a*d)*f*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*Ellip
ticF[I*ArcSinh[Sqrt[-(b/a)]*x], (a*d)/(b*c)])/(Sqrt[-(b/a)]*c*d*(-(b*c) + a*d)*Sqrt[a - b*x^2]*Sqrt[c - d*x^2]
)

Maple [A] (verified)

Time = 4.95 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.40

method	result	size
default	\(\frac {\left (-\sqrt {\frac {d}{c}}\, b c f \,x^{3}-\sqrt {\frac {d}{c}}\, b d e \,x^{3}+\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, F\left (x \sqrt {\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right ) a d e -\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, F\left (x \sqrt {\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right ) b c e -\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, E\left (x \sqrt {\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right ) a c f -\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, E\left (x \sqrt {\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right ) a d e +\sqrt {\frac {d}{c}}\, a c f x +\sqrt {\frac {d}{c}}\, a d e x \right ) \sqrt {-b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}}{\sqrt {\frac {d}{c}}\, c \left (a d -b c \right ) \left (b d \,x^{4}-a d \,x^{2}-c b \,x^{2}+a c \right )}\)	\(338\)
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}\, \left (-\frac {\left (b d \,x^{2}-a d \right ) x \left (c f +d e \right )}{d c \left (a d -b c \right ) \sqrt {\left (x^{2}-\frac {c}{d}\right ) \left (b d \,x^{2}-a d \right )}}+\frac {\left (-\frac {f}{d}+\frac {c f +d e}{d c}-\frac {a \left (c f +d e \right )}{c \left (a d -b c \right )}\right ) \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, F\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {b d \,x^{4}-a d \,x^{2}-c b \,x^{2}+a c}}+\frac {\left (c f +d e \right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \left (F\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )-E\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )\right )}{\left (a d -b c \right ) c \sqrt {\frac {d}{c}}\, \sqrt {b d \,x^{4}-a d \,x^{2}-c b \,x^{2}+a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}}\)	\(384\)

[In]

int((f*x^2+e)/(-d*x^2+c)^(3/2)/(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

(-(d/c)^(1/2)*b*c*f*x^3-(d/c)^(1/2)*b*d*e*x^3+((-d*x^2+c)/c)^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticF(x*(d/c)^(1/2
),(b*c/a/d)^(1/2))*a*d*e-((-d*x^2+c)/c)^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticF(x*(d/c)^(1/2),(b*c/a/d)^(1/2))*b*
c*e-((-d*x^2+c)/c)^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticE(x*(d/c)^(1/2),(b*c/a/d)^(1/2))*a*c*f-((-d*x^2+c)/c)^(1
/2)*((-b*x^2+a)/a)^(1/2)*EllipticE(x*(d/c)^(1/2),(b*c/a/d)^(1/2))*a*d*e+(d/c)^(1/2)*a*c*f*x+(d/c)^(1/2)*a*d*e*
x)*(-b*x^2+a)^(1/2)*(-d*x^2+c)^(1/2)/(d/c)^(1/2)/c/(a*d-b*c)/(b*d*x^4-a*d*x^2-b*c*x^2+a*c)

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.06 \[ \int \frac {e+f x^2}{\sqrt {a-b x^2} \left (c-d x^2\right )^{3/2}} \, dx=-\frac {{\left (a c d^{2} e + a c^{2} d f\right )} \sqrt {-b x^{2} + a} \sqrt {-d x^{2} + c} x - {\left (a c d^{2} e + a c^{2} d f - {\left (a d^{3} e + a c d^{2} f\right )} x^{2}\right )} \sqrt {a c} \sqrt {\frac {d}{c}} E(\arcsin \left (x \sqrt {\frac {d}{c}}\right )\,|\,\frac {b c}{a d}) + {\left ({\left ({\left (b c^{2} d - a d^{3}\right )} e + {\left (a c^{2} d - a c d^{2}\right )} f\right )} x^{2} - {\left (b c^{3} - a c d^{2}\right )} e - {\left (a c^{3} - a c^{2} d\right )} f\right )} \sqrt {a c} \sqrt {\frac {d}{c}} F(\arcsin \left (x \sqrt {\frac {d}{c}}\right )\,|\,\frac {b c}{a d})}{a b c^{4} d - a^{2} c^{3} d^{2} - {\left (a b c^{3} d^{2} - a^{2} c^{2} d^{3}\right )} x^{2}} \]

[In]

integrate((f*x^2+e)/(-d*x^2+c)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

-((a*c*d^2*e + a*c^2*d*f)*sqrt(-b*x^2 + a)*sqrt(-d*x^2 + c)*x - (a*c*d^2*e + a*c^2*d*f - (a*d^3*e + a*c*d^2*f)
*x^2)*sqrt(a*c)*sqrt(d/c)*elliptic_e(arcsin(x*sqrt(d/c)), b*c/(a*d)) + (((b*c^2*d - a*d^3)*e + (a*c^2*d - a*c*
d^2)*f)*x^2 - (b*c^3 - a*c*d^2)*e - (a*c^3 - a*c^2*d)*f)*sqrt(a*c)*sqrt(d/c)*elliptic_f(arcsin(x*sqrt(d/c)), b
*c/(a*d)))/(a*b*c^4*d - a^2*c^3*d^2 - (a*b*c^3*d^2 - a^2*c^2*d^3)*x^2)

Sympy [F]

\[ \int \frac {e+f x^2}{\sqrt {a-b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {e + f x^{2}}{\sqrt {a - b x^{2}} \left (c - d x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((f*x**2+e)/(-d*x**2+c)**(3/2)/(-b*x**2+a)**(1/2),x)

[Out]

Integral((e + f*x**2)/(sqrt(a - b*x**2)*(c - d*x**2)**(3/2)), x)

Maxima [F]

\[ \int \frac {e+f x^2}{\sqrt {a-b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {f x^{2} + e}{\sqrt {-b x^{2} + a} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((f*x^2+e)/(-d*x^2+c)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

integrate((f*x^2 + e)/(sqrt(-b*x^2 + a)*(-d*x^2 + c)^(3/2)), x)

Giac [F]

\[ \int \frac {e+f x^2}{\sqrt {a-b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {f x^{2} + e}{\sqrt {-b x^{2} + a} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((f*x^2+e)/(-d*x^2+c)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

integrate((f*x^2 + e)/(sqrt(-b*x^2 + a)*(-d*x^2 + c)^(3/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {e+f x^2}{\sqrt {a-b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {f\,x^2+e}{\sqrt {a-b\,x^2}\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \]

[In]

int((e + f*x^2)/((a - b*x^2)^(1/2)*(c - d*x^2)^(3/2)),x)

[Out]

int((e + f*x^2)/((a - b*x^2)^(1/2)*(c - d*x^2)^(3/2)), x)